Augmented matrix: Definition from Answers.com

In linear algebra, the augmented matrix of a matrix is obtained by changing a matrix in some way.

Given the matrices A and B, where:

$A = \begin{bmatrix} 1 & 3 & 2 \\ 2 & 0 & 1 \\ 5 & 2 & 2 \end{bmatrix} , \quad B = \begin{bmatrix} 4 \\ 3 \\ 1 \end{bmatrix}.$

Then, the augmented matrix (A|B) is written as:

$(A|B)= \left[\begin{array}{ccc|c} 1 & 3 & 2 & 4 \\ 2 & 0 & 1 & 3 \\ 5 & 2 & 2 & 1 \end{array}\right].$

This is useful when solving systems of linear equations or the augmented matrix may also be used to find the inverse of a matrix by combining it with the identity matrix.

Examples

Let C be a square 2×2 matrix where $C = \begin{bmatrix} 1 & 3 \\ -5 & 0 \end{bmatrix}.$

To find the inverse of C we create (C|I) where I is the 2×2 identity matrix. We then reduce the part of (C|I) corresponding to C to the identity matrix using only elementary matrix transformations on (C|I).

$(C|I) = \left[\begin{array}{cc|cc} 1 & 3 & 1 & 0\\ -5 & 0 & 0 & 1 \end{array}\right]$

$(I|C^{-1}) = \left[\begin{array}{cc|cc} 1 & 0 & 0 & -\frac{1}{5} \\ 0 & 1 & \frac{1}{3} & \frac{1}{15} \end{array}\right]$

As used in linear algebra, an augmented matrix is used to represent the coefficients and the solution vector of each equation set. For the set of equations:

$\begin{align} x_1 + 2x_2 + 3x_3 &= 0 \\ 3x_1 + 4x_2 + 7x_3 &= 2 \\ 6x_1 + 5x_2 + 9x_3 &= 11 \end{align}$

the augmented matrix would be composed of

$A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 7 \\ 6 & 5 & 9 \end{bmatrix} , \quad B = \begin{bmatrix} 0 \\ 2 \\ 11 \end{bmatrix}.$

Leaving us with:

$(A|B) = \left[\begin{array}{ccc|c} 1 & 2 & 3 & 0 \\ 3 & 4 & 7 & 2 \\ 6 & 5 & 9 & 11 \end{array}\right],$ or

$(A|B) = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ \end{array}\right].$

References

Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Dover Publications, 1992, ISBN 0-486-67102-X. Page 31.

External links

Code to display an Augumented Matrix

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